ABCF->ab-angle a

Average Error: 52.3 → 43.9

Time: 54.4s

Precision: binary64

Cost: 15072

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

↓

\[\begin{array}{l} t_0 := \frac{1}{-\frac{\frac{{B}^{2} - A \cdot \left(C \cdot 4\right)}{C}}{\sqrt{F \cdot \left(A \cdot -16\right)}}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -2.85 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-140}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-286}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+252}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (-
           (/
            (/ (- (pow B 2.0) (* A (* C 4.0))) C)
            (sqrt (* F (* A -16.0)))))))
        (t_1 (* 0.25 (sqrt (* (/ F C) -16.0)))))
   (if (<= B -1.9e-30)
     (- (sqrt (* -2.0 (/ F B))))
     (if (<= B -2.85e-111)
       t_0
       (if (<= B -4e-140)
         (- (sqrt (- (/ F C))))
         (if (<= B -1.05e-245)
           t_1
           (if (<= B -5.5e-286)
             (- (sqrt (- (/ F A))))
             (if (<= B -6.5e-296)
               (* 0.25 (sqrt (* (/ F A) -16.0)))
               (if (<= B 6.6e-307)
                 t_1
                 (if (<= B 1.85e-106)
                   t_0
                   (if (<= B 5.4e+252)
                     (* (/ (sqrt 2.0) B) (- (sqrt (* F B))))
                     (- (sqrt (* (/ F B) 2.0))))))))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = 1.0 / -(((pow(B, 2.0) - (A * (C * 4.0))) / C) / sqrt((F * (A * -16.0))));
	double t_1 = 0.25 * sqrt(((F / C) * -16.0));
	double tmp;
	if (B <= -1.9e-30) {
		tmp = -sqrt((-2.0 * (F / B)));
	} else if (B <= -2.85e-111) {
		tmp = t_0;
	} else if (B <= -4e-140) {
		tmp = -sqrt(-(F / C));
	} else if (B <= -1.05e-245) {
		tmp = t_1;
	} else if (B <= -5.5e-286) {
		tmp = -sqrt(-(F / A));
	} else if (B <= -6.5e-296) {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	} else if (B <= 6.6e-307) {
		tmp = t_1;
	} else if (B <= 1.85e-106) {
		tmp = t_0;
	} else if (B <= 5.4e+252) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * B));
	} else {
		tmp = -sqrt(((F / B) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 * (((b ** 2.0d0) - ((4.0d0 * a) * c)) * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / ((b ** 2.0d0) - ((4.0d0 * a) * c))
end function

↓

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / -((((b ** 2.0d0) - (a * (c * 4.0d0))) / c) / sqrt((f * (a * (-16.0d0)))))
    t_1 = 0.25d0 * sqrt(((f / c) * (-16.0d0)))
    if (b <= (-1.9d-30)) then
        tmp = -sqrt(((-2.0d0) * (f / b)))
    else if (b <= (-2.85d-111)) then
        tmp = t_0
    else if (b <= (-4d-140)) then
        tmp = -sqrt(-(f / c))
    else if (b <= (-1.05d-245)) then
        tmp = t_1
    else if (b <= (-5.5d-286)) then
        tmp = -sqrt(-(f / a))
    else if (b <= (-6.5d-296)) then
        tmp = 0.25d0 * sqrt(((f / a) * (-16.0d0)))
    else if (b <= 6.6d-307) then
        tmp = t_1
    else if (b <= 1.85d-106) then
        tmp = t_0
    else if (b <= 5.4d+252) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * b))
    else
        tmp = -sqrt(((f / b) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	return -Math.sqrt(((2.0 * ((Math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / (Math.pow(B, 2.0) - ((4.0 * A) * C));
}

↓

public static double code(double A, double B, double C, double F) {
	double t_0 = 1.0 / -(((Math.pow(B, 2.0) - (A * (C * 4.0))) / C) / Math.sqrt((F * (A * -16.0))));
	double t_1 = 0.25 * Math.sqrt(((F / C) * -16.0));
	double tmp;
	if (B <= -1.9e-30) {
		tmp = -Math.sqrt((-2.0 * (F / B)));
	} else if (B <= -2.85e-111) {
		tmp = t_0;
	} else if (B <= -4e-140) {
		tmp = -Math.sqrt(-(F / C));
	} else if (B <= -1.05e-245) {
		tmp = t_1;
	} else if (B <= -5.5e-286) {
		tmp = -Math.sqrt(-(F / A));
	} else if (B <= -6.5e-296) {
		tmp = 0.25 * Math.sqrt(((F / A) * -16.0));
	} else if (B <= 6.6e-307) {
		tmp = t_1;
	} else if (B <= 1.85e-106) {
		tmp = t_0;
	} else if (B <= 5.4e+252) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * B));
	} else {
		tmp = -Math.sqrt(((F / B) * 2.0));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	return -math.sqrt(((2.0 * ((math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / (math.pow(B, 2.0) - ((4.0 * A) * C))

↓

def code(A, B, C, F):
	t_0 = 1.0 / -(((math.pow(B, 2.0) - (A * (C * 4.0))) / C) / math.sqrt((F * (A * -16.0))))
	t_1 = 0.25 * math.sqrt(((F / C) * -16.0))
	tmp = 0
	if B <= -1.9e-30:
		tmp = -math.sqrt((-2.0 * (F / B)))
	elif B <= -2.85e-111:
		tmp = t_0
	elif B <= -4e-140:
		tmp = -math.sqrt(-(F / C))
	elif B <= -1.05e-245:
		tmp = t_1
	elif B <= -5.5e-286:
		tmp = -math.sqrt(-(F / A))
	elif B <= -6.5e-296:
		tmp = 0.25 * math.sqrt(((F / A) * -16.0))
	elif B <= 6.6e-307:
		tmp = t_1
	elif B <= 1.85e-106:
		tmp = t_0
	elif B <= 5.4e+252:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * B))
	else:
		tmp = -math.sqrt(((F / B) * 2.0))
	return tmp

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	t_0 = Float64(1.0 / Float64(-Float64(Float64(Float64((B ^ 2.0) - Float64(A * Float64(C * 4.0))) / C) / sqrt(Float64(F * Float64(A * -16.0))))))
	t_1 = Float64(0.25 * sqrt(Float64(Float64(F / C) * -16.0)))
	tmp = 0.0
	if (B <= -1.9e-30)
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B))));
	elseif (B <= -2.85e-111)
		tmp = t_0;
	elseif (B <= -4e-140)
		tmp = Float64(-sqrt(Float64(-Float64(F / C))));
	elseif (B <= -1.05e-245)
		tmp = t_1;
	elseif (B <= -5.5e-286)
		tmp = Float64(-sqrt(Float64(-Float64(F / A))));
	elseif (B <= -6.5e-296)
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	elseif (B <= 6.6e-307)
		tmp = t_1;
	elseif (B <= 1.85e-106)
		tmp = t_0;
	elseif (B <= 5.4e+252)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * B))));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp = code(A, B, C, F)
	tmp = -sqrt(((2.0 * (((B ^ 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / ((B ^ 2.0) - ((4.0 * A) * C));
end

↓

function tmp_2 = code(A, B, C, F)
	t_0 = 1.0 / -((((B ^ 2.0) - (A * (C * 4.0))) / C) / sqrt((F * (A * -16.0))));
	t_1 = 0.25 * sqrt(((F / C) * -16.0));
	tmp = 0.0;
	if (B <= -1.9e-30)
		tmp = -sqrt((-2.0 * (F / B)));
	elseif (B <= -2.85e-111)
		tmp = t_0;
	elseif (B <= -4e-140)
		tmp = -sqrt(-(F / C));
	elseif (B <= -1.05e-245)
		tmp = t_1;
	elseif (B <= -5.5e-286)
		tmp = -sqrt(-(F / A));
	elseif (B <= -6.5e-296)
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	elseif (B <= 6.6e-307)
		tmp = t_1;
	elseif (B <= 1.85e-106)
		tmp = t_0;
	elseif (B <= 5.4e+252)
		tmp = (sqrt(2.0) / B) * -sqrt((F * B));
	else
		tmp = -sqrt(((F / B) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(1.0 / (-N[(N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(A * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] / N[Sqrt[N[(F * N[(A * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[Sqrt[N[(N[(F / C), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.9e-30], (-N[Sqrt[N[(-2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B, -2.85e-111], t$95$0, If[LessEqual[B, -4e-140], (-N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision]), If[LessEqual[B, -1.05e-245], t$95$1, If[LessEqual[B, -5.5e-286], (-N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision]), If[LessEqual[B, -6.5e-296], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.6e-307], t$95$1, If[LessEqual[B, 1.85e-106], t$95$0, If[LessEqual[B, 5.4e+252], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(N[(F / B), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if B < -1.9000000000000002e-30

   1. Initial program 53.7

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 53.6

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 53.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 53.7	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around -inf 55.2

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{-1 \cdot B}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]

   4. Simplified 55.2

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{\left(-B\right)}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]
      Proof

      [Start] 55.2	\[ \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + -1 \cdot B\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 55.2	\[ \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{B \cdot -1}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-9 [=>] 55.2	\[ \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{\left(-B\right)}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]

   5. Taylor expanded in C around 0 63.0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\right)} \]

   6. Simplified 63.0

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)} \]
      Proof

      [Start] 63.0	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\right) \]
      rational.json-simplify-43 [=>] 63.0	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F \cdot \left(A - B\right)} \cdot -1\right)} \]
      rational.json-simplify-2 [<=] 63.0	\[ \frac{\sqrt{2}}{B} \cdot \left(\sqrt{\color{blue}{\left(A - B\right) \cdot F}} \cdot -1\right) \]
      rational.json-simplify-9 [=>] 63.0	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A - B\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - B\right)}}\right) \]

   7. Taylor expanded in A around 0 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{\frac{F}{B}}\right)} \]

   8. Simplified 37.9

      \[\leadsto \color{blue}{-\sqrt{-2 \cdot \frac{F}{B}}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{\frac{F}{B}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{\frac{F}{B}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{\frac{F}{B}}} \]
      rational.json-simplify-2 [=>] 64.0	\[ -\color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\sqrt{2} \cdot \sqrt{-1}\right)} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{-1 \cdot 2}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\frac{F}{B}} \cdot \sqrt{\color{blue}{-2}} \]
      exponential.json-simplify-20 [=>] 37.9	\[ -\color{blue}{\sqrt{-2 \cdot \frac{F}{B}}} \]

   if -1.9000000000000002e-30 < B < -2.85e-111 or 6.59999999999999999e-307 < B < 1.8499999999999999e-106

   1. Initial program 51.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 50.1

      \[\leadsto \color{blue}{\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 51.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 51.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 51.0	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 51.0	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Applied egg-rr 50.4

      \[\leadsto \color{blue}{\frac{1}{A \cdot \left(C \cdot 4\right) - {B}^{2}} \cdot \sqrt{\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(F + F\right)\right)}} \]

   4. Applied egg-rr 50.3

      \[\leadsto \color{blue}{\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{2 \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right)}}}} \]

   5. Taylor expanded in C around inf 64.0

      \[\leadsto \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\color{blue}{\sqrt{A \cdot F} \cdot \left(\sqrt{2} \cdot \left(C \cdot \sqrt{-8}\right)\right)}}} \]

   6. Simplified 51.6

      \[\leadsto \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\color{blue}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}} \]
      Proof

      [Start] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{A \cdot F} \cdot \left(\sqrt{2} \cdot \left(C \cdot \sqrt{-8}\right)\right)}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{A \cdot F} \cdot \color{blue}{\left(C \cdot \left(\sqrt{-8} \cdot \sqrt{2}\right)\right)}}} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{A \cdot F} \cdot \left(C \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{-8}\right)}\right)}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\color{blue}{C \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{A \cdot F}\right)}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{A \cdot F}\right)}} \]
      exponential.json-simplify-20 [=>] 51.6	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \color{blue}{\sqrt{\left(A \cdot F\right) \cdot \left(-8 \cdot 2\right)}}}} \]
      rational.json-simplify-2 [=>] 51.6	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\color{blue}{\left(F \cdot A\right)} \cdot \left(-8 \cdot 2\right)}}} \]
      metadata-eval [=>] 51.6	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\left(F \cdot A\right) \cdot \color{blue}{-16}}}} \]

   7. Applied egg-rr 51.0

      \[\leadsto \frac{1}{\color{blue}{-\frac{\frac{{B}^{2} - A \cdot \left(C \cdot 4\right)}{C}}{\sqrt{F \cdot \left(A \cdot -16\right)}}}} \]

   if -2.85e-111 < B < -3.9999999999999999e-140

   1. Initial program 47.9

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 48.4

      \[\leadsto \color{blue}{\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 47.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 47.9	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 47.9	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 47.9	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right)} \]

   4. Simplified 53.2

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{C}}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{C}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{C}} \]
      exponential.json-simplify-20 [=>] 53.2	\[ -\color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
      rational.json-simplify-9 [=>] 53.2	\[ -\sqrt{\color{blue}{-\frac{F}{C}}} \]

   if -3.9999999999999999e-140 < B < -1.05000000000000005e-245 or -6.49999999999999963e-296 < B < 6.59999999999999999e-307

   1. Initial program 51.9

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 52.5

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 51.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 51.9	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{C}}\right)} \]

   4. Simplified 50.5

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}} \]
      Proof

      [Start] 64.0	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.25 \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{\frac{F}{C}}\right) \]
      metadata-eval [=>] 64.0	\[ 0.25 \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{\frac{F}{C}}\right) \]
      exponential.json-simplify-20 [=>] 50.5	\[ 0.25 \cdot \color{blue}{\sqrt{\frac{F}{C} \cdot -16}} \]

   if -1.05000000000000005e-245 < B < -5.4999999999999998e-286

   1. Initial program 54.1

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 53.7

      \[\leadsto \color{blue}{\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 54.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 54.1	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 54.1	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 54.1	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in C around inf 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right)} \]

   4. Simplified 49.5

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{A}}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{A}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{A}} \]
      exponential.json-simplify-20 [=>] 49.5	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 49.5	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   if -5.4999999999999998e-286 < B < -6.49999999999999963e-296

   1. Initial program 55.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 53.9

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 55.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 55.8	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in C around inf 64.0

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{A}}\right)} \]

   4. Simplified 50.1

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}} \]
      Proof

      [Start] 64.0	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.25 \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{\frac{F}{A}}\right) \]
      metadata-eval [=>] 64.0	\[ 0.25 \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{\frac{F}{A}}\right) \]
      exponential.json-simplify-20 [=>] 50.1	\[ 0.25 \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}} \]

   if 1.8499999999999999e-106 < B < 5.40000000000000021e252

   1. Initial program 50.4

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 50.3

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 50.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 50.4	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around inf 54.7

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{\left(0.5 \cdot \frac{{\left(A - C\right)}^{2}}{B} + B\right)}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]

   4. Taylor expanded in C around 0 43.8

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right)} \]

   5. Simplified 43.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(0.5 \cdot \frac{{A}^{2}}{B} + \left(B + A\right)\right)}\right)} \]
      Proof

      [Start] 43.8	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 43.8	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 43.8	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 43.8	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right)}}\right) \]
      rational.json-simplify-41 [=>] 43.8	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \color{blue}{\left(0.5 \cdot \frac{{A}^{2}}{B} + \left(B + A\right)\right)}}\right) \]

   6. Taylor expanded in A around 0 42.6

      \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{\sqrt{F \cdot B}}\right) \]

   if 5.40000000000000021e252 < B

   1. Initial program 64.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around inf 64.0

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{\left(0.5 \cdot \frac{{\left(A - C\right)}^{2}}{B} + B\right)}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]

   4. Taylor expanded in C around 0 38.8

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right)} \]

   5. Simplified 38.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(0.5 \cdot \frac{{A}^{2}}{B} + \left(B + A\right)\right)}\right)} \]
      Proof

      [Start] 38.8	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 38.8	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 38.8	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 38.8	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right)}}\right) \]
      rational.json-simplify-41 [=>] 38.8	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \color{blue}{\left(0.5 \cdot \frac{{A}^{2}}{B} + \left(B + A\right)\right)}}\right) \]

   6. Taylor expanded in A around 0 28.5

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   7. Simplified 28.3

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
      Proof

      [Start] 28.5	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      rational.json-simplify-2 [=>] 28.5	\[ \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 28.5	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      exponential.json-simplify-20 [=>] 28.3	\[ -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

3. Recombined 8 regimes into one program.

4. Final simplification 43.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -2.85 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{-\frac{\frac{{B}^{2} - A \cdot \left(C \cdot 4\right)}{C}}{\sqrt{F \cdot \left(A \cdot -16\right)}}}\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-140}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-286}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-307}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{-\frac{\frac{{B}^{2} - A \cdot \left(C \cdot 4\right)}{C}}{\sqrt{F \cdot \left(A \cdot -16\right)}}}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+252}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Error	43.5
Cost	40972

\[\begin{array}{l} t_0 := \frac{1}{-\frac{\frac{{B}^{2} - A \cdot \left(C \cdot 4\right)}{C}}{\sqrt{F \cdot \left(A \cdot -16\right)}}}\\ t_1 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_2 := A \cdot \left(4 \cdot C\right)\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-29}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - t_2\right) \cdot F\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_2 - {B}^{2}}\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-176}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-189}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 2
Error	43.5
Cost	40908

\[\begin{array}{l} t_0 := \frac{1}{-\frac{\frac{{B}^{2} - A \cdot \left(C \cdot 4\right)}{C}}{\sqrt{F \cdot \left(A \cdot -16\right)}}}\\ t_1 := A \cdot \left(4 \cdot C\right)\\ t_2 := t_1 - {B}^{2}\\ t_3 := \left({B}^{2} - t_1\right) \cdot F\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{-36}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-177}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-188}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;B \leq 1.06 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 5.7 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 3
Error	43.7
Cost	21324

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := A \cdot \left(4 \cdot C\right)\\ t_2 := \sqrt{F \cdot \left(A \cdot -16\right)}\\ t_3 := \frac{1}{-\frac{\frac{{B}^{2} - t_0}{C}}{t_2}}\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-38}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-114}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - t_1\right) \cdot F\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{C}}{t_2} \cdot \left(t_0 - {B}^{2}\right)}\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-233}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-105}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{+252}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 4
Error	43.5
Cost	13708

\[\begin{array}{l} \mathbf{if}\;B \leq -2.8 \cdot 10^{-41}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\frac{A \cdot \left(C \cdot 4\right)}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 5
Error	43.6
Cost	7640

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{-19}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{-211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-252}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 6
Error	43.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{-30}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\frac{A \cdot \left(C \cdot 4\right)}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 7
Error	43.5
Cost	7508

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;B \leq -2.85 \cdot 10^{-18}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -9.2 \cdot 10^{-208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-250}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-65}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 8
Error	43.7
Cost	7444

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-18}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 9
Error	49.4
Cost	7116

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;B \leq -1.06 \cdot 10^{-18}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-248}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	47.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;C \leq -1.3 \cdot 10^{-294}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 11
Error	54.8
Cost	6720

\[-\sqrt{-\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023067 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))